= Three-to-one mass double-pendulum modes
{title2=$\omega_-^2=2g/(3\ell),\quad\omega_+^2=2g/\ell$}
An upper <mass> $3m$ and lower <mass> $m$ joined by equal <light> strings of length $\ell$ have small-angle <mass> and <stiffness matrices> $m\ell^2\left(\begin{smallmatrix}4&1\\1&1\end{smallmatrix}\right)$ and $mg\ell\left(\begin{smallmatrix}4&0\\0&1\end{smallmatrix}\right)$. The squared <frequencies> are $2g/(3\ell)$ and $2g/\ell$, with angular eigenvectors $(1,2)$ and $(1,-2)$ respectively. Equal-angle release decomposes with coefficients $3\alpha/4$ and $\alpha/4$. The angles are measured from vertical, not between strings.
Back to article page
